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Properties

Label 2-198-11.8-c2-0-3
Degree $2$
Conductor $198$
Sign $-0.0413 - 0.999i$
Analytic cond. $5.39510$
Root an. cond. $2.32273$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.831 + 1.14i)2-s + (−0.618 + 1.90i)4-s + (5.40 + 3.92i)5-s + (−2.30 − 0.748i)7-s + (−2.68 + 0.874i)8-s + 9.45i·10-s + (6.21 + 9.07i)11-s + (−0.0214 − 0.0295i)13-s + (−1.05 − 3.25i)14-s + (−3.23 − 2.35i)16-s + (−8.00 + 11.0i)17-s + (20.4 − 6.63i)19-s + (−10.8 + 7.85i)20-s + (−5.22 + 14.6i)22-s − 36.7·23-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (−0.154 + 0.475i)4-s + (1.08 + 0.785i)5-s + (−0.329 − 0.106i)7-s + (−0.336 + 0.109i)8-s + 0.945i·10-s + (0.564 + 0.825i)11-s + (−0.00165 − 0.00227i)13-s + (−0.0756 − 0.232i)14-s + (−0.202 − 0.146i)16-s + (−0.470 + 0.648i)17-s + (1.07 − 0.349i)19-s + (−0.540 + 0.392i)20-s + (−0.237 + 0.666i)22-s − 1.59·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0413 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0413 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-0.0413 - 0.999i$
Analytic conductor: \(5.39510\)
Root analytic conductor: \(2.32273\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1),\ -0.0413 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.43897 + 1.49970i\)
\(L(\frac12)\) \(\approx\) \(1.43897 + 1.49970i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.831 - 1.14i)T \)
3 \( 1 \)
11 \( 1 + (-6.21 - 9.07i)T \)
good5 \( 1 + (-5.40 - 3.92i)T + (7.72 + 23.7i)T^{2} \)
7 \( 1 + (2.30 + 0.748i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (0.0214 + 0.0295i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (8.00 - 11.0i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-20.4 + 6.63i)T + (292. - 212. i)T^{2} \)
23 \( 1 + 36.7T + 529T^{2} \)
29 \( 1 + (-45.1 - 14.6i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-5.03 + 3.65i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (8.70 - 26.7i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-34.5 + 11.2i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 + 84.6iT - 1.84e3T^{2} \)
47 \( 1 + (25.5 + 78.7i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-24.0 + 17.4i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (-5.43 + 16.7i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-39.8 + 54.8i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 74.0T + 4.48e3T^{2} \)
71 \( 1 + (-5.47 - 3.98i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-91.3 - 29.6i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (27.5 + 37.9i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (11.8 - 16.3i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 142.T + 7.92e3T^{2} \)
97 \( 1 + (97.2 - 70.6i)T + (2.90e3 - 8.94e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.64692849003027614383253199291, −11.70676994684529289412274690061, −10.27281804971174662523545104999, −9.713053921824758629392427967460, −8.402987800045104154868166617785, −6.96017008628135014235527386624, −6.42849950550472654869479859503, −5.24171136733067028949865813967, −3.75344747926467011937622456244, −2.20527884626587724221174298522, 1.18567742843824782828532905273, 2.78239337536429630682876895254, 4.36218230412792540188612111470, 5.61697406308153245766681542293, 6.36285421998118649058736159753, 8.189337559578415898521303731775, 9.406887017764963009861254036467, 9.818128053860249673521974476064, 11.17745923895348858793297459331, 12.10061750936216804556351921432

Graph of the $Z$-function along the critical line